Problem 852

Let $d_n=p_{n+1}-p_n$ , where $p_n$ is the $n$ th prime. Let $h(x)$ be maximal such that for some $n<x$ the numbers $d_n,d_{n+1},\ldots,d_{n+h(x)-1}$ are all distinct. Estimate $h(x)$ . In particular, is it true that

h(x) >(\log x)^c

for some constant $c>0$ , and

h(x)=o(\log x)?

Source: erdosproblems.com/852

Number theory
Primes

Wiki page

Status

Open

No claim settles this problem.

Open. The site's label is OPEN, and its commentary records only that Brun's sieve gives $h(x)\to\infty$ . Two partial claims posted on the problem's discussion thread on 24 April 2026, Chojecki's bound $h(x)\gg(\log x)^{1/3}$ (credited to GPT-5.5 Pro) and Turturean's four-prime count (made with a scaffold on ChatGPT-5.5-Pro), each claim the lower bound $h(x)\gg(\log x)^{1/3}$ , which answers the first particular question with yes for every fixed $c<1/3$ . Both are claimed and unreviewed, and neither estimates $h(x)$ or decides whether $h(x)=o(\log x)$ , so the problem stays open.

Problem 852

Status

Proof claims

Discussion